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Question Number 82160 by M±th+et£s last updated on 18/Feb/20

$$provethat$$$$\underset{x\to \mathrm{\infty }}{lim}{n}^2\sqrt{(1-cos\left(\frac{1}{n}\right)\sqrt{(1-cos\frac{1}{n})\sqrt{(1-cos\frac{1}{n})...}}}=\frac{1}{2}$$

Answered by MJS last updated on 18/Feb/20

$$w=\sqrt{(1-\cos \frac{1}{n})\sqrt{(1-\cos \frac{1}{n})\sqrt{...}}}$$$$w^2=(1-\cos \frac{1}{n})w\Rightarrow w=0\vee w=1-\cos \frac{1}{n}$$$$...=\underset{n\to \mathrm{\infty }}{\lim }{n}^2(1-\cos \frac{1}{n})$$$$\mathrm{let}n=\frac{1}{k}$$$$\underset{k\to 0}{\lim }\frac{1-\cos k}{k^2}=\underset{k\to 0}{\lim }\frac{\frac{d^2}{{dk}^2}[1-\cos k]}{\frac{d^2}{{dk}^2}\left[k^2\right]}=$$$$=\underset{k\to 0}{\lim }\frac{\cos k}{2}=\frac{1}{2}$$

Answered by mr W last updated on 18/Feb/20

$$1-\cos \frac{1}{n}=2{\sin }^2\frac{1}{2n}$$$$\underset{n\to \mathrm{\infty }}{lim}{n}^2\sqrt{(1-cos\left(\frac{1}{n}\right)\sqrt{(1-cos\frac{1}{n})\sqrt{(1-cos\frac{1}{n})...}}}$$$$=\underset{n\to \mathrm{\infty }}{lim}{n}^2{\left(2{\sin }^2\frac{1}{2n}\right)}^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...}$$$$=\underset{n\to \mathrm{\infty }}{lim}{n}^2{\left(2{\sin }^2\frac{1}{2n}\right)}^{\frac{1}{2}\times \frac{1}{1-\frac{1}{2}}}$$$$=\underset{n\to \mathrm{\infty }}{lim}{n}^2\left(2{\sin }^2\frac{1}{2n}\right)$$$$=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{2}{\left(\frac{\sin \frac{1}{2n}}{\frac{1}{2n}}\right)}^2$$$$=\frac{1}{2}\times 1=\frac{1}{2}$$

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